Symbolic Powers of Defining Ideals of Veronese Rings Fangu Chen - PowerPoint PPT Presentation

Symbolic Powers of Defining Ideals of Veronese Rings Fangu Chen joint work with Alan Tang REU summer 2019 at University of Michigan supervised by Professors Eric Canton, Elo´ ısa Grifo, Jack Jeffries 1

Overview Ideals and Varieties Primary Decomposition Symbolic Powers Veronese Ring and Ideal Our Results 2

Ideals and Varieties

Variety Example Let f ∈ R = R [ x , y , z ] defined by f ( x , y , z ) = 2 x − y + 3 z . Then V ( f ) := { ( x , y , z ) ∈ R 3 | f ( x , y , z ) = 0 } is a plane. 3

Variety Example Let f ∈ R = R [ x , y , z ] defined by f ( x , y , z ) = 2 x − y + 3 z . Then V ( f ) := { ( x , y , z ) ∈ R 3 | f ( x , y , z ) = 0 } is a plane. Definition Let R = C [ x 1 , . . . , x n ] be a polynomial ring with C - coefficients and variables x 1 , . . . , x n . A variety is the set of common zeroes in C n of a collection of polynomials f i ∈ R . The variety associated to the set { f 1 , . . . , f m } is written V ( f 1 , . . . , f m ). 3

Ideals and Varieties If p is a point in the variety, then p is also a zero of any polynomial combination m � g i ( x 1 , . . . , x n ) f i ( x 1 , . . . , x n ) i =1 4

Ideals and Varieties If p is a point in the variety, then p is also a zero of any polynomial combination m � g i ( x 1 , . . . , x n ) f i ( x 1 , . . . , x n ) i =1 Observation The variety V ( f 1 , . . . , f m ) depends only on the ideal I generated by { f 1 , . . . , f m } . 4

Ideals and Varieties Given a variety of an ideal J , we can form the set I ( V ( J )) of all polynomials vanishing on this variety. It is easy to check the set is an ideal. 5

Ideals and Varieties Given a variety of an ideal J , we can form the set I ( V ( J )) of all polynomials vanishing on this variety. It is easy to check the set is an ideal. If f ∈ J , then by definition f ( p ) = 0 ∀ p ∈ V ( J ). Hence, f ∈ I ( V ( J )), so J ⊆ I ( V ( J )). 5

Ideals and Varieties Given a variety of an ideal J , we can form the set I ( V ( J )) of all polynomials vanishing on this variety. It is easy to check the set is an ideal. If f ∈ J , then by definition f ( p ) = 0 ∀ p ∈ V ( J ). Hence, f ∈ I ( V ( J )), so J ⊆ I ( V ( J )). If J ⊆ C [ x 1 , . . . , x n ] is radical, then J = I ( V ( J )). Definition Given an ideal I in a ring R , the radical of I is √ I = { f ∈ R : f n ∈ I for some n ∈ N } . √ An ideal I is radical if I = I . 5

Primary Decomposition

Ideals and Varieties If I , J in a polynomial ring R , then V ( I ∩ J ) = V ( I ) ∪ V ( J ). 6

Ideals and Varieties If I , J in a polynomial ring R , then V ( I ∩ J ) = V ( I ) ∪ V ( J ). Example Consider the ideal I = ( xz , yz ) = ( z ) ∩ ( x , y ) in R [ x , y , z ]. In R 3 , V ( z ) corresponds to the xy -plane and V ( x , y ) corresponds to the z -axis, then V ( I ) = V ( z ) ∪ V ( x , y ). 6

Primary Decomposition Given an ideal, we would like to decompose it as an intersection of simpler ideals. 7

Primary Decomposition Given an ideal, we would like to decompose it as an intersection of simpler ideals. Example In the ring of integers Z , suppose a positive integer n has prime factorization n = p a 1 1 . . . p a r r , then the ideal ( n ) = ( p a 1 1 ) ∩ · · · ∩ ( p a r r ). For example, for 60 = 2 2 · 3 · 5, we have (60) = (4) ∩ (3) ∩ (5). 7

Primary Decomposition Definition An ideal Q � = (1) in a ring R is primary if fg ∈ Q implies either f ∈ Q or g m ∈ Q for some m ∈ N . 8

Primary Decomposition Definition An ideal Q � = (1) in a ring R is primary if fg ∈ Q implies either f ∈ Q or g m ∈ Q for some m ∈ N . Definition A primary decomposition of an ideal I consists of primary ideals Q 1 , . . . , Q n such that I = ∩ n i =1 Q i . A primary decomposition I = ∩ n i =1 Q i is irredundant if ∩ i � = j Q i � = I for each j ∈ { 1 , . . . , n } and √ Q i � = � Q j for all i � = j . 8

Primary Decomposition Definition An ideal Q � = (1) in a ring R is primary if fg ∈ Q implies either f ∈ Q or g m ∈ Q for some m ∈ N . Definition A primary decomposition of an ideal I consists of primary ideals Q 1 , . . . , Q n such that I = ∩ n i =1 Q i . A primary decomposition I = ∩ n i =1 Q i is irredundant if ∩ i � = j Q i � = I for each j ∈ { 1 , . . . , n } and √ Q i � = � Q j for all i � = j . Theorem Any ideal in a Noetherian ring has a primary decomposition. 8

Symbolic Powers

Symbolic Powers C [ x ] • f vanishes at p • ( x − p ) | f or f ∈ ( x − p ) x = p is a root of f • ( x − p ) k | f or f ∈ ( x − p ) k • f vanishes to x = p is a root of f , f ′ , . . . , f ( k − 1) order ≥ k at p 9

Symbolic Powers C [ x ] • f vanishes at p • ( x − p ) | f or f ∈ ( x − p ) x = p is a root of f • ( x − p ) k | f or f ∈ ( x − p ) k • f vanishes to x = p is a root of f , f ′ , . . . , f ( k − 1) order ≥ k at p C [ x 1 , . . . , x n ] • f ∈ ( x 1 − p 1 , . . . , x n − p n ) k • f vanishes to x = ( p 1 , . . . , p n ) is a root of order ≥ k at ∂ d 1 . . . ∂ dn n f for all d 1 + · · · + d n < k ( p 1 , . . . , p n ) ∂ x d 1 ∂ x dn 1 9

Symbolic Powers Definition (Zariski-Nagata) Let R = C [ x 1 , . . . , x n ] and I a radical ideal in R . Then the k-th symbolic power of I is I ( k ) = { f ∈ R | f vanishes to order ≥ k at every x ∈ V ( I ) } . 10

Symbolic Powers Definition (Zariski-Nagata) Let R = C [ x 1 , . . . , x n ] and I a radical ideal in R . Then the k-th symbolic power of I is I ( k ) = { f ∈ R | f vanishes to order ≥ k at every x ∈ V ( I ) } . I ( k ) = { f ∈ R | ∂ d 1 . . . ∂ dn n f ∈ I for all d 1 + · · · + d n < k } ∂ x d 1 ∂ x dn 1 10

Symbolic Powers Definition (Zariski-Nagata) Let R = C [ x 1 , . . . , x n ] and I a radical ideal in R . Then the k-th symbolic power of I is I ( k ) = { f ∈ R | f vanishes to order ≥ k at every x ∈ V ( I ) } . I ( k ) = { f ∈ R | ∂ d 1 . . . ∂ dn n f ∈ I for all d 1 + · · · + d n < k } ∂ x d 1 ∂ x dn 1 • C n • C [ x 1 , . . . , x n ] • radical ideal I • vanish on variety V ( I ) • symbolic power I ( k ) • vanish to order k over variety 10

Veronese Ring and Ideal

Veronese Ring and Ideal Example Let S 2 = C [ x 1 , x 2 ]. 11

Veronese Ring and Ideal Example Let S 2 = C [ x 1 , x 2 ]. The 3-rd Veronese subring S 2 , 3 := ( C [ x 1 , x 2 ]) 3 = C [ x 3 1 , x 2 1 x 2 , x 1 x 2 2 , x 3 2 ]. 11

Veronese Ring and Ideal Example Let S 2 = C [ x 1 , x 2 ]. The 3-rd Veronese subring S 2 , 3 := ( C [ x 1 , x 2 ]) 3 = C [ x 3 1 , x 2 1 x 2 , x 1 x 2 2 , x 3 2 ]. Let R = C [ t 1 , t 2 , t 3 , t 4 ]. 11

Veronese Ring and Ideal Example Let S 2 = C [ x 1 , x 2 ]. The 3-rd Veronese subring S 2 , 3 := ( C [ x 1 , x 2 ]) 3 = C [ x 3 1 , x 2 1 x 2 , x 1 x 2 2 , x 3 2 ]. Let R = C [ t 1 , t 2 , t 3 , t 4 ]. There is a surjective ring homomorphism π : R → S 2 , 3 defined by t 1 �→ x 3 1 , t 2 �→ x 2 1 x 2 , t 3 �→ x 1 x 2 2 , t 4 �→ x 3 2 , and I 2 , 3 := ker( π ). 11

Veronese Ring and Ideal Example Let S 2 = C [ x 1 , x 2 ]. The 3-rd Veronese subring S 2 , 3 := ( C [ x 1 , x 2 ]) 3 = C [ x 3 1 , x 2 1 x 2 , x 1 x 2 2 , x 3 2 ]. Let R = C [ t 1 , t 2 , t 3 , t 4 ]. There is a surjective ring homomorphism π : R → S 2 , 3 defined by t 1 �→ x 3 1 , t 2 �→ x 2 1 x 2 , t 3 �→ x 1 x 2 2 , t 4 �→ x 3 2 , and I 2 , 3 := ker( π ). Note that π ( t 1 t 3 − t 2 2 ) = x 4 1 x 2 2 − x 4 1 x 2 2 = 0 π ( t 1 t 4 − t 2 t 3 ) = x 3 1 x 3 2 − x 3 1 x 3 2 = 0 π ( t 2 t 4 − t 2 3 ) = x 2 1 x 4 2 − x 2 1 x 4 2 = 0 11

Veronese Ring and Ideal Example Let S 2 = C [ x 1 , x 2 ]. The 3-rd Veronese subring S 2 , 3 := ( C [ x 1 , x 2 ]) 3 = C [ x 3 1 , x 2 1 x 2 , x 1 x 2 2 , x 3 2 ]. Let R = C [ t 1 , t 2 , t 3 , t 4 ]. There is a surjective ring homomorphism π : R → S 2 , 3 defined by t 1 �→ x 3 1 , t 2 �→ x 2 1 x 2 , t 3 �→ x 1 x 2 2 , t 4 �→ x 3 2 , and I 2 , 3 := ker( π ). Note that π ( t 1 t 3 − t 2 2 ) = x 4 1 x 2 2 − x 4 1 x 2 2 = 0 π ( t 1 t 4 − t 2 t 3 ) = x 3 1 x 3 2 − x 3 1 x 3 2 = 0 π ( t 2 t 4 − t 2 3 ) = x 2 1 x 4 2 − x 2 1 x 4 2 = 0 In fact, I 2 , 3 = ker( π ) = ( t 1 t 3 − t 2 2 , t 1 t 4 − t 2 t 3 , t 2 t 4 − t 2 3 ), and → V ( t 1 t 3 − t 2 2 , t 1 t 4 − t 2 t 3 , t 2 t 4 − t 2 3 ) ⊂ C 4 . I 2 , 3 ← 11

Veronese Ring and Ideal Definition Let S n = C [ x 1 , . . . , x n ]. The d -th Veronese in n variables 1 , x d − 1 S n , d := ( C [ x 1 , . . . , x n ]) d = C [ x d x 2 , . . . , x d n ]. Let 1 R = k [ t 1 , t 2 , . . . , t ( n + d − 1 )]. There is a surjective ring d homomorphism − → R S n , d x d t 1 �− → 1 x d − 1 �− → t 2 x 2 1 . . . x d �− → t ( n + d − 1 ) n d We write I n , d for the kernel of this map. 12

Our Results

� n + d − 2 � The ideal I n , d ( n , d ≥ 2) is generated by 2-minors of a n × d − 1 matrix. 13

� n + d − 2 � The ideal I n , d ( n , d ≥ 2) is generated by 2-minors of a n × d − 1 matrix. Example � � t (3 , 0) t (2 , 1) t (1 , 2) where t (3 , 0) �→ x 3 1 , t (2 , 1) �→ x 2 I 2 , 3 = I 2 1 x 2 , t (2 , 1) t (1 , 2) t (0 , 3) t (1 , 2) �→ x 1 x 2 2 , t (0 , 3) �→ x 3 2 . 13

Example � � t (4 , 0) t (3 , 1) t (2 , 2) t (1 , 3) I 2 , 4 = I 2 . t (3 , 1) t (2 , 2) t (1 , 3) t (0 , 4) 14